All of the 4th grade teachers and students from Santa Rita went on a field trip to an archaeology museum. Tickets were $$5.50$ each for teachers and $$4.50$ each for students, and the group paid $$57.00$ in total. The next month, the same group visited a natural history museum where the tickets cost $$16.50$ each for teachers and $$12.00$ each for students, and the group paid $$157.50$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5.5x+4.5y = 57}$ ${16.5x+12y = 157.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-16.5x-13.5y = -171}$ ${16.5x+12y = 157.5}$ Add the top and bottom equations together. $ -1.5y = -13.5 $ $ y = \dfrac{-13.5}{-1.5}$ ${y = 9}$ Now that you know ${y = 9}$ , plug it back into $ {5.5x+4.5y = 57}$ to find $x$ ${5.5x + 4.5}{(9)}{= 57}$ $5.5x+40.5 = 57$ $5.5x = 16.5$ $x = \dfrac{16.5}{5.5}$ ${x = 3}$ You can also plug ${y = 9}$ into $ {16.5x+12y = 157.5}$ and get the same answer for $x$ ${16.5x + 12}{(9)}{= 157.5}$ ${x = 3}$ There were $3$ teachers and $9$ students on the field trips.